Matchgate and space-bounded quantum computations are equivalent

TL;DR

This paper demonstrates that matchgate quantum circuits are computationally equivalent to space-bounded quantum computations with logarithmic space, linking a special class of quantum gates to a broader complexity framework.

Contribution

It establishes a formal equivalence between matchgate circuit power and logarithmic-space bounded quantum computation, clarifying their computational capabilities.

Findings

Matchgate circuits are equivalent to logarithmic-space bounded quantum computations.

Polynomial-sized matchgate circuits are classically simulatable.

Matchgate circuits' power coincides with polynomial-time, logarithmic-space bounded quantum computation.

Abstract

Matchgates are an especially multiflorous class of two-qubit nearest neighbour quantum gates, defined by a set of algebraic constraints. They occur for example in the theory of perfect matchings of graphs, non-interacting fermions, and one-dimensional spin chains. We show that the computational power of circuits of matchgates is equivalent to that of space-bounded quantum computation with unitary gates, with space restricted to being logarithmic in the width of the matchgate circuit. In particular, for the conventional setting of polynomial-sized (logarithmic-space generated) families of matchgate circuits, known to be classically simulatable, we characterise their power as coinciding with polynomial-time and logarithmic-space bounded universal unitary quantum computation.